Problem: $ D = \left[\begin{array}{rr}4 & -2 \\ 2 & 4 \\ 2 & 1\end{array}\right]$ $ F = \left[\begin{array}{rr}1 & 4 \\ 1 & -1\end{array}\right]$ Is $ D F$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ D$ , have? How many rows does the second matrix, $ F$ , have? Since $ D$ has the same number of columns (2) as $ F$ has rows (2), $ D F$ is defined.